It was shown in [1]} that, for any scalar \(\psi \)  in LRS spacetime, the dot and hat derivatives satisfy the relation
\(\Omega \dot{\psi }=\xi \hat{\psi }.\) 
This relation was used to obtain the following explicit algebraic expressions for \(\phi \)  and \(Q\) , obtained through the employment of the commutation relation (REF ):
\(\phi &=\frac{\Omega }{\xi }\left(\frac{2}{3}\Theta -\Sigma \right),\\Q&=-\frac{\Omega \xi }{\left(\Omega ^2+\xi ^2\right)}\left(\rho +p+\Pi \right).\) 


Let \(H\)  be an marginally trapped tube in \(M\)  foliated by marginally trapped surfaces. We will show that if either one of \(\Omega \)  and \(\xi \)  is non-zero on \(H\) , then either \(\left(2/3\right)\Theta -\Sigma =0\Rightarrow \phi =0\)  or \(\phi =0\Rightarrow \left(2/3\right)\Theta -\Sigma =0\) , in which case \(H\)  is minimal.
Suppose we have that \(\Omega =0\)  and \(\xi \ne 0\) . Then, from (REF ) we have that \(\phi =0\) , in which case we must have \(\left(2/3\right)\Theta -\Sigma =0\) . Hence, \(H\)  is minimal. On the other hand, if \(\Omega \ne 0\)  and \(\xi =0\) , then (REF ) gives
\(\Omega \left(\frac{2}{3}\Theta -\Sigma \right)=0.\) 
Since \(\Omega \ne 0\) , we must have \(\left(2/3\right)\Theta -\Sigma =0\) , in which case \(\phi =0\) . Hence \(H\)  is also minimal.
Now, consider the case in which \(\Omega \ne 0\)  and \(\xi \ne 0\) . Then, on the horizon (REF ) becomes (using (REF ))
\(\phi \left(\frac{\Omega }{\xi }+1\right)=0.\) 
We know that \(\phi \ne 0\) , since otherwise \(H\)  would be minimal. Therefore we must instead have that
\(\frac{\Omega }{\xi }=-1.\) 
We also know that, in LRS spacetimes with simultaneous rotation and spatial twist, the existence of homothetic Killing vectors in the plane spanned by the unit tangent direction and the preferred spatial direction is guaranteed by Theorem \(1.\)  of [1]}. Therefore \(H\)  also admits a homothetic Killing vector. The condition for \(H\)  to be null is that \(\Omega /\xi =\pm 1\) , clearly satisfied by \(H\)  according to (REF ), which implies that \(H\)  is a Killing horizon (see the reference [1]} for more discussion). However, since \(H\)  is null, one must have \(C=0\) , which, from (REF ) gives
\(2\left(\xi ^2+2\xi \Omega -\Omega ^2\right)-\left(\rho +p+\Pi \right)+2Q=0\) 
Substituting (REF ) into (REF ) and () and simplifying, (REF ) reduces to \(\xi ^2=0\Rightarrow \xi =0\)  (or \(\Omega ^2=0\Rightarrow \Omega =0\) ). The cases \(\xi =0\)  and \(\Omega =0\)  have already been shown to yield a minimal \(H\) . The result therefore follows.\(\Box \)
Discussion
The result of Theorem REF  severely constrains the spacetimes with locally rotational symmetry that can admit dynamical horizons and timelike membranes. In particular, Theorem REF  tells us that, for spacetimes with locally rotational symmetry where all vector and tensor quantities vanish identically, rotation and spatial twist are obstacles to the existence of truly dynamical black holes. Hence, only the class II of these spacetimes with \(\Omega =0\)  and \(\xi =0\)  admit such horizons. Well known examples of this class include the Lemaitre-Tolman-Bondi (LTB) solution, Schwarzschild spacetime (although the MTT here, a null horizon, is minimal since both expansion scalars are identically zero), the Oppenheimer-Snyder (OS) dust solution and the Robertson-Walker (RW) solution. (The LTB and RW solutions, for example, contain dynamical horizons foliated by non-minimal MOTS, and the OS dust collapse contains timelike membranes, also foliated by nonminimal MOTS. (These were recently studied in context of the 1+1+2 formulation [4]}.)
While this itself does not expound on physical application of the obtained result, it certainly adds to the growing literature on the existence problem for dynamical horizons in particular.
In the numerical context, the coordinate dependence of (REF ) already restricts the possible solutions in the LRS class of spacetimes that could be useful in modelling mergers. The result of this work appears to verify this, and rules out many LRS solutions which could possibly be used to model black hole mergers. Two important subclasses are the spatially homogeneous LRS III class of spacetimes with \(\xi \ne 0\)  and \(\Omega \)  and the LRS I solutions with \(\xi =0\)  and \(\Omega \ne 0\) . A known example of the latter is the Gödel rotating solution.
An immediate problem to consider would be the rigidity of the result of this paper, under perturbations, where one has an LRS background.

Acknowledgements
Gratitude is expressed to the anonymous referee who provided invaluable comments and suggestions that have enhanced the clarity of the result of this paper. The author acknowledges support by the First Rand Bank, through the Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa.
